Problem: Solve for $x$ and $y$ using elimination. $\begin{align*}2x+4y &= -6 \\ -7x-5y &= 8\end{align*}$
Solution: We can eliminate $y$ when its corresponding coefficients are negative inverses. Recalling our knowledge of least common multiples, multiply the top equation by $5$ and the bottom equation by $4$ $\begin{align*}10x+20y &= -30\\ -28x-20y &= 32\end{align*}$ Add the top and bottom equations. $-18x = 2$ Divide both sides by $-18$ and reduce as necessary. $x = -\dfrac{1}{9}$ Substitute $-\dfrac{1}{9}$ for $x$ in the top equation. $2( -\dfrac{1}{9})+4y = -6$ $-\dfrac{2}{9}+4y = -6$ $4y = -\dfrac{52}{9}$ $y = -\dfrac{13}{9}$ The solution is $\enspace x = -\dfrac{1}{9}, \enspace y = -\dfrac{13}{9}$.